POLYMER SOLUTIONS - WordPress.com · 2013-12-10 · NllU ELSEVIER Fluid Phase Equilibria 104 (1995) 229-252 CONTINUOUS THERMODYNAMICS FOR POLYDISPERSE POLYMER SOLUTIONS Ying Hu a,

NllU ELSEVIER Fluid Phase Equilibria 104 (1995) 229-252

CONTINUOUS THERMODYNAMICS FOR POLYDISPERSE POLYMER SOLUTIONS

Ying Hu a, Xugen yinga, D. T. Wu b and J. M. Prausnitz c

aThermodynamics Research Laboratory, East China University of Science and Technology, Shanghai 200237 (China)

bMarshall Laboratory, E. I. Du Pont de Nemours & Company, Philadelphia, PA 19146 (USA)

CDepartment of Chemical Engineering, University of California, Berkeley and Chemical Sciences Division, Lawrence Berkeley Laboratory, Berkeley, CA 94720 (USA) Keywords: theory, application, liquid-liquid equilibria, polydisperse polymers

Received 12 January 1994; accepted in final form 20 June 1994

ABSTRACT

Continuous thermodynamics is a framework which combines continuum modeling for the compositions of complex and multicomponent mixtures with molecular thermodynamic models and efficient numerical methods.

In this work, a generalized molecular-thermodynamic model for polydisperse polymer solutions is developed; it is formally similar to the classical Flory-Huggins theory but with a polymer-size dependent and polymer-concentration dependent Flory parameter. Most existing lattice models and equation-of-state models such as the Guggenheim, Orifino-Flory, Koningsveld-Kleintjens, Sanchez-Lacombe and Revised Freed models can be cast into this generalized model but with different polymer-size and polymer-concentration dependence for the Flory parameter. A generalized continuous-thermodynamics framework based on this generalized model is also presented; expressions for chemical potentials, spinodals and critical points are derived using both the discrete multicomponent method and the continuous functional procedure. Internally consistent results are obtained. Criteria for multiple critical points are also derived. Computer programs are established for polydisperse-polymer solutions with either a standard or an arbitrary distribution for the polymer's molecular weight; in the latter case, the derivative method is applied, based on a previously developed spline fit. To illustrate the framework developed here, calculated liquid-liquid-equilibrium phase diagrams are shown, including UCST, LCST and hour-glass-shaped cloud-point curves, shadow curves, spinodals, critical points and their dependence on molecular parameters, on pressure and on molecular-weight distribution properties.

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230 K Hu et al. / Fluid Phase Equilibria 104 (1995) 229-252

INTRODUCTION

Typical industrial polymers are polydisperse. Like the compositions of other complex mixtures such as petroleum, shale oil, coal-derived liquids and vegetable oils, those of polymer solutions or polymer blends are not given by a few discrete values of concentration but, instead, by a continuous distribution function. Because polydisperse polymer solutions contain a very large number of components, accurate determinations of concentrations of all components are impractical. Continuous thermodynamics provides a framework that replaces discrete compositions with a continuous distribution function. Cotterman and Prausnitz (1991) have given an extensive review concerning application of continuous thermodynamics to chemical process design.

Polydispersity is often not important for vapor-liquid equilibria in polymer- solvent mixtures but it is often very important for liquid-liquid equilibria. Spino- dais, critical points and higher-order critical points play an important role. A useful continuous-thermodynamic framework should therefore include criteria for stability of various orders. Two different procedures are found in the literature. The first one, the discrete multicomponent procedure, considers polymer species as discrete components. Expressions for chemical potentials, spinodals and critical points are derived for a multicomponent system with very many components while the continuous distribution function for polymer species is only used for calculations of various moments that appear in those expressions. Koningsveld et al. (Koningsveld and Staverman, 1968; Koningsveld, Kleintjens and Schultz, 1970) have presented a thorough analysis for this procedure and developed general calculation methods for liquid-liquid equilibria of polydisperse polymer solutions using a generalized Flory-Huggins model where the Flory parameter is composition-dependent. Those methods were extended by ~olc (1970, 1975) to investigate the effect of the distribution function on calculated cloud-point curves. Flory and Frost (1978) used essentially the same procedure to describe phase separation in polydisperse systems. ~olc, Kleintjens and Koningsveld (1984) studied the stability problem for finding multiple critical points by using a series-expansion method.

The second and more recent procedure is the functional method developed primarily by Rfitzsch and coworkers (Kehlen, R~itzsch and Bergmann, 1987; Beer- baum, Bergmann, Kehlen and R~itzsch, 1987; Browarzik, Kehlen, R~itzsch and Bergmann, 1990; Bergmann, Teichert, Kehlen and R~itzsch, 1992). In their framework, thermodynamic functions were defined for continuous mixtures. Expressions for chemical potentials, spinodals and critical points were derived using functional theory, where high-order variations were obtained by using the Lagrange method of undetermined multipliers for minimizing the second-order differential of the Gibbs energy. Results were compared with those by the discrete method using the simple Flory-Huggins model. Although the functional method is attractive due to its mathematical integrity, it is still in a stage of development. Its reliability should often be checked with results by the discrete approach because the monomer, the fundamental unit of a polymer, has a finite size; therefore, a continuous distribution function for the description of composition is at best a good approximation.

Y. Hu et al. /F lu id Phase Equilibria 104 (1995) 229-252 231

Although the continuous-thermodynamic methods published in the literature can qualitatively describe the cloud-point curve and the shadow curve which express relations between equlibrium temperature and compositions for principal phase and conjugate phase, respectively, for a polydisperse-polymer solution, these methods are not accurate enough because the molecular-thermodynamic model used is too simple. It has been known for many years that the classical Flory-Huggins theory for a close-packed lattice cannot describe phase diagrams exhibiting lower critical solution temperatures (LCST) or hour-glass shaped coexistence curves. Even for systems exhibiting only upper critical solution temperatures (UCST), the Flory-Huggins parameter is not constant but depends on concentration and some- times, on the polymer's molar mass. Theoretical improvements have been obtained by various workers, including Guggenheim (1952), Orifino and Flory (1957) and Koningsveld, Kleintjens and Schultz (1971) who improved the mathematical description of the lattice model. Other improvements were achieved by considering the volume change during mixing which is neglected in lattice theory. Flory et al. (Flory, Orwoll and Vrij, 1964; Flory, 1970), Patterson and Delmas (1970), Beret and Prausnitz (1975) and Donohue et al. (Donohue and Prausnitz, 1978; Ikonomou and Donohue, 1986) developed an equation-of-state theory which introduced a free-volume contribution into the partition function. Sanchez and Lacombe (1978) and Kleintjens and Koningsveld (1980) developed a lattice-fluid theory by introducing holes into the Flory-Huggins lattice. All those theories can describe semiquanti- tatively LCST, UCST and hour-glass shaped coexistence curves.

Freed and coworkers (Bewendi and Freed, 1988; Dudowicz and Freed, 1991) developed a new lattice-cluster theory for polymer systems which is formally an exact mathematical solution of the Flory-Huggins lattice. Based on Freed theory, we have developed a close-packed lattice model (Hu, Lambert, Soane and Prausnitz, 1991) and a lattice-fluid model (Hu, Ying, Wu and Prausnitz, 1993), which give consistent coexistence curves in comparison with those calculated by the rigorous Freed theory when r 1=1 and r2=l-10,000 , and with computer simula- tion results by Madden, Pesci and Freed (1988) when r l= l and r2-=100, where r refers to chain length. The models have been applied to real polymer solutions with UCST, LCST and hour-glass phase diagrams as well as to reproducing the pressure dependence and the molar-mass dependence of liquid-liquid spinodals and binodals.

In this work, we first introduce a generalized molecular-thermodynamic model for polydisperse polymer solutions which is formally the same as the Flory- Huggins theory but which contains a specified composition-dependent and chain- length-dependent Flory parameter. Most models can be cast into this generalized model by using different expressions for the composition dependence and the chain-length dependence of the Flory parameter. A generalized continuous- thermodynamic framework based on this generalized model is then developed; expressions for chemical potentials, spinodals and critical points of various orders are derived using the discrete method. Correspoding derivations using the functional method are given in the Appendix. Results from the two methods are shown to be consistent. Calculated liquid-liquid equilibria under various conditions are presented to illustrate the proposed framework.

232 Y. Hu et al. / Fluid Phase Equilibria 104 (1995) 229-252

GENERALIZED MOLECULAR-THERMODYNAMIC MODEL FOR BINARY POLYDISPERSE POLYMER SOLUTIONS

N r k T

For a mixture containing one solvent and one polydisperse polymer, we adopt the following expressions for the Helmholtz energy of mixing AmixA and the Gibbs energy of mixing Am/x G :

Amix A OP o dP i = - - l n c I ) o + ] ~ 7 . . lnqbi + dPo]~dPig ? (1)

ro i i i

Ami x G ~ o dPi ** - - - - l n c b o + ] ~ - - l n ~ i + (I~o~_,tl)ig i (2)

Gv N r k Z ro i ri i

where "4v and Gv are the reduced Helmholtz energy and the reduced Gibbs energy of mixing per site; N r is the total number of sites in the lattice; subscripts o and i stands for the solvent and the polymer solute i; r o , r i and ~ o , dPi are corresponding chain lengths and volume fractions; ~s mer denoted by subscript s.

OP o = N oro / N r , dP i = N iri / N r

N r = N o r o + ~]Vir i i

is the volume fraction of the total poly-

]~qb i--cD s --- 1 -c I ) o (3) i

(4)

In eqs.(1) and (2), gi and gi are Flow-parameter functions for AmixA and AmixG ,

respectively. For close-packed lattice models, AmixA = A m i x G , gi = gi • Gen- erally, gi depends on the temperature, the pressure, the chain length r i and the totolume fraction of the polymer ~ s , gi = gi ( T , P , ri , cD s ).

Now we show relations between this generalized model and various existing models.

C a s e 1. F l o r y - H u g g i n s M o d e l

For polydisperse polymer solutions, The Flory-Huggins theory is given by

Ami x G Ami x A ~ o OP i . . . . . . lntD o + ]~-77-. ln<bi + OPodPsg(FH) (5) N r k T N r k T r o i i

where the Flory-Huggins parameter g(FH) is independent of the chain length and the polymer volume fraction. Comparing with eqs.(1) and (2), we have

• * * g (FH) (6) gi = gi ="

C a s e 2. G u g g e n h e i m M o d e l

For polydisperse polymer solutions, Guggenheim's theory (1952) gives the following Helmholtz energy of mixing:

Y. Hu et al. / Fluid Phase Equilibria 104 (1995) 229-252 233

Ami x A ~ o dPi _ _ _- - - l n ~ ° + ~ - - l n ~ i N r k T ro i ri

dp Oo z dp qi ln (9i dp ° ~.dp igi (Gu) (7) + -2z o q ° l n - - + - - , ~ i 4" r o dP o 2 i ri ~ i i

where energy parameter gi (Gu) can be chain-length dependent; z is the coordination number; qo, Oo and qi, Oi are molecular parameters characterizing molecular sur- face areas and corresponding area fractions for the solvent o and the polymer solute i , respectively.

Oo = Noqo / Nq , Oi = Niqi / Nq (8)

Nq -- N o qo + ~ V i qi (9) i

Parameters qo and qi are defined by

z - 2 2 z - 2 2 z - 2 q® ~- ro + - - , qi = r i - - + - -- r i - (10)

Z Z Z Z Z

When r i is large, 2/z is negligible, justifying eq.(lO).

Substituting eqs(8-10) into eq.(7), after rearrangement, we have

AmixA ¢bo ~ i z_~ q® lnq° z q i lnqi --N rkT = - - l n O ° r o + ~--~.. l n ~ i i i + 2 O--r® r® + ~ i ~ i ri ri

z (dpo --+q° ~dpi q i ) l n [ 1 - 2 ( 1 - c b ° ) ] + ~ o E ~ i g i (Gu) (11) 2 ro 7 r i z r o i

Comparing with generalized eqs.(1) and (2), we have

q® in q®

qi qi gi = gi = gi (Gu) + 2 ( ** * + In )

ro ~ s ro ri ~ o ri

z ( qo qi __alP°)] _ _ _ _ + ) 1 n [ 1 - 2 ( 1 - (12)

2 r o • s r i • o z r o

Case 3. Orif ino-Flory Model and Koningsveld-Kleintjens Model

Orifino and Flory (1957) revised the Flory-Huggins equation by replacing the volume fraction tI~ i in the energy term with the area fraction ®i. Koningsveld and Kleintjens (1971) further revised the Orifino-Flory model by adding another entropy term which contains a parameter or. For polydisperse polymer solutions, eq.(5) is rewitten:

Ami x A d~ o t~ i . . . . l n~ o + ,~_~--ln~i + fi)o~,dPiG~i(ri) + ¢I)o~.~®igi (KK) (13) N r k T r® i r i i i

where the entropy parameter (x i and the energy parameter gi (KK) can be chain- length dependent. For the Orifino-Flory model, txi---0.

234 K Hu et al. / F h d d Phase Equilibria 104 (1995) 229-252

Substituting eqs.(8) and (9) into eq.(13) and comparing with generalized eqs.(1) and (2), we have

** • qi 2 ( l_dPO )l_lgi(KK ) gi = gi ~- °t(ri ) + [1- (14) r i Z F o

Case 4. R e v i s e d F r e e d M o d e l

The Flory-Huggins parameter for the revised Freed model (Hu, Lambert, Soane and Prausnitz, 1991) is

g**i = gi* ~- g ( R F ) . = 192 ( rol r~-I )2 + 2~ i

+ @s + - - ( 1 - @ s ) - 1.611~2@s(1-qbs) (15) r o r i

where ~i is a reduced energy parameter, ~i = e i / k T . Parameter ~i depends on the chain length.

e i = e e + e r r i - d (16)

where e e and e r are two characteristic constants; d is a positive number.

For practical reason, it is necessary to introduce effective chain lengths ro (e)

and ri (e) for the solvent and the polymer solute, and corresponding effective volume fractions ~o (e) and ~(e) to make the model more flexible especially for fitting critical coordinates. Because of the changing environment, effective chain lengths are composition-dependent given by

r ( e ) = r o ( l + c r ~ s ) , ri(e) = r i ( l -CrdPo) (17)

where c r is a binary size parameter. The total number of sites in the lattice remains unchanged.

N r = N o r o + ~ N i r i = Noro (e) + ~ I i r i (e) (18) i i

cI) (o e) = ~i)o (14-¢ r CI)s ) , OiJ ( e ) = ~ i ( 1 - C r ~ o ) (19)

Eq.(1) is rewritten: ~.(e)

*°(e) ln*o (e) + ~ - e ) lnqb~ e ) + - ' " o . i i Amix A _ ~ ( e ) ~ (e)g (RF)

N r kT ro (e) i ri i

~i) o dP i di) o - l n O o + ] ~ - - lnOi + In(1 +c r Os )

ro i r i ro

+]~ ~! ln(1-c r di) o )-I-di) o •di) i (1 + c r fI) s ) ( 1 - c r di) o )gi (RF ) (20) i ri i

where gi (RF) is expressed by eq.(15) with ro, r i and d) s replaced by ro (e), ri (e) a n d

qbs(e), respectively.


Comparing with eqs.(1) and (2), we have

** • 1 l n ( l + c r ~ ) + 1 ln(1-CrdOo) + (l+CrCI~,.)(1-Cr~o)gi (RF) (21) gi = gi - gO s r ~ " dPori

Case 5. Lat t ice-Fluid Mode l

We have developed a lattice-fluid model (Hu, Ying, Wu and Prausnitz, 1993) based on Case 4 using a two-step approach. The expression for the Helmholtz energy of mixing is

Ami x A Ami x A a Ami x A b - - - + ( 2 2 ) N r kT N r kT N r kT

where subscripts a and b denote contributions by the close-packed lattice and by introducing holes, respectively. For the former, we use eq.(20) of Case 4. For the latter,

1 0 ~ - n r ) m - i A m i x A b _ 1 -~) ln(1-~) + ln~) + ~]~a,n n (23) N r k T p ru m n

The corresponding equation of state is

p = ;? [ - ln(1-~) - ~(1-r,71) + ]~]~(m -1)a° , , T-" ~9'" ] (24) 111 #l

where /5 , 0 and T are reduced pressure, reduced density and reduced temperature, respectively, defined by

= pv-e ' /gb , [3 = N r V - ~ / V , T = k T / e b (25)

where v-* is the hard-core volume of one segment, t b is an energy parameter of the mixture defined by the following mixing rule:

2 % = ~ o too + 2 ~ o ~ s %s + ~s2~ss (26)

where too, %s and tss are interaction energies for solvent-solvent, solvent-solute and solute-solute pairs, respectively. We neglect here the subtle effect on energies due to different chain lengths of solute species. The number-average chain length of the mixture r u in eqs.(23) and (24) is defined by

r u 1 = dO o r(71 + ~.adPi ri -1 -- O o ro --1 (27) i

Here ri -1 is neglected because the chain length of the polymer solute is sufficiently 0 in eqs.(23) and (24) long in comparison with that of the solvent. Coefficients a,, m

are functions of r u given in [25].

If we take a 1°1 = 3 and a° l = -3 , other amn are all zero, eq.(24) is reduced to the Sanchez-Lacombe equation of state (1978).

Because of the mixing rules used in defining t b and ru, AmixA b of eq.(23) and /3 of eq.(24) depend only on the total volume fraction qbs, not on the individual volume fractions q~i- The Helmholtz energy of mixing of eq.(22) can therefore be compared with the generalized eq.(1). We then have


gi = gai + AmixAb / dOodOsNr kT (28)

where gai is the contribution by the close-packed lattice expressed by eq.(21). In that contribution, energy parameter e i should be treated as chain-length independent for consistency, and be replaced by eoo+ess-2~os. The second term in eq.(28) is given by eq.(23).

The Gibbs energy of mixing can be written as

Ami x G Ami x A A(p V ) - - - + - ( 2 9 )

N r kT N r kT N r kT

Because the contribution to the pressure by the close-packing state is zero, A ( p V ) / N r k T =/%~-1~-1, which also depends only on the total volume fraction of the polymer. Comparing with the generalized eq.(2), we then have

gi = gi +/3]~-1[ )-1 ] dOo dOs (30)

with gi given by eq.(28).

CHEMICAL POTENTIALS AND PHASE-EQUILIBRIUM CALCULATIONS

Expressions for chemical potentials of solvent o and polymer solute i are derived directly from the generalized eq.(2).

(go -go*)/kT = ~(Amix G IkT) / ON o :¢:g **1

= ln(1--~s) + dOs(1-ro/rn) + rodOs]~[dOjg j - (1-dOs) dOjgj ] J

(~t i -}.ti"V)/kT = ~(AmixG [kT) [ ~N i

--- lndo/ + 1 - ri /r o + r idos(1-ro /rn) / r o + (1-dos)r ig i

- ri(1-dos)~.~[dOjg j - (1-dOs) dOjg;* ' ] J

(31)

(32)

where gj = (~gj / 3dOs)T,p is the chemical potential for the reference state which is the close-packed pure component at system temperature. The number-

is calculated by average chain length of the polymer solute denoted by r n

r ~ 1 ---- ]~dOi ri-l[dOs • i

When two phases (cz) and ([3) are at equilibrium,

We define

Lo = lndOo + dOs (1-r o/r n ) + r o dO2sS o * * * * I

So = ]F,(gk -doo gt )@t ~dOs , k

Substitution into eqs.(31),(32) and (33) yields

(33)

(34)

(35)


InK i = In -- ri{[dPog.~-rollndPo--dPsSo] (a) - [Oog*-ro l lndPo-dPsSo] (~)} (36) ~/(c0

where K i = ~(f)) [ ¢}(a) is the partition coefficient for component i.

Now we establish two fundamental equations for phase-equilibrium calculations for binary solutions with a polydisperse polymer solute.

F o = 1 - L ( a ) / L o (B) - -O , F s = 1-ModO(a)t~(f~)s ' s - - 0 (37)

where M o = ~.J~idPi(a)/cb(s cO = dP(~)/dP(a)s s . By definition, M0~(a)/~(13)s , s is unity. i

The function F o of eq.(37) is for the solvent, while the function F s is for the polymer; it contains information concerning all discrete polymer components. By solv- ing the two equations in eq.(37), we can obtain cloud-point curves and shadow curves.

STABILITY CRITERIA OF VARIOUS ORDERS

Multiple critical points arise as multiple nontrivial roots of phase-equilibrium equations that happen to coincide with the critical composition (~olc, Kleintjens and Koningsveld, 1984). Their significance is in their association with multiphase equilibria. For example, a triple critical point [i.e. a tricritical point] can be viewed as a terminus of a three-phase line where the three coexistence phases become criti- cally identical. Solutions of polydisperse polymers open new possibilities for study- ing multiphase equilibria because of their large number of components and their large number of degrees of freedom. Although experimental results are rare at present for multiple critical points except for spinodals and ordinary critical points, a systematic study of the conditions of multiple critical phenomena is scientifically interesting and perhaps helpful for finding and using such phenomena.

The spinodal criterion and the critical-point criteria are well-known as shown in the theory of stability in many textbooks, for example (Modell and Reid, 1983). We extend them to obtain higher-order stability criteria. Generally, the stability of m th order can be established by the following determinants:

I - (~-1) D(~C-1) D~:-I) D2 . . .

= " ~ G21 G22 • • . G 2 n = 0 ~: = m , m - 1 . . . . . 2 ( 3 8 ) D(~) /

L Gnl Gn2 " ' ' Gnn

where n is the total number of polymer components which approaches infinity.

~2Gv ~D 0¢-1) G i j = ~)~pi~)dpj , Dk ('c-1)= ~)<b k (39)

For closure, we further define D~ 1) - - G l t or D (1) ~ ~Gv / ~ 1 . When m=2, eq.(38) gives the criterion for spinodal; m=3, for critical point; m=4, for double critical point; rn =5, for triple critical point; etc.

238 Y. Hu et al. /Fhdd Phase Equilibria 104 (1995) 229-252

We now adopt the generalized model established above. Substituting eq.(2) for Gv into eq.(39), we have

Oi i 1 + 1 ** = 2g i - 2Zqbkg/~*' + (1-dos)(2g/**: + £qbkg/~*" ) (40) r o (1-dO s ) ri doi k k

Oi j = 1 ** ** 2 - **" **" **' ~'do **" r o ( l _ ~ s ) gi - g j - ~_flPkgk + ( 1 - ~ s ) ( g i + g j + 2.d k g k )(41)

k k

**,, "~2 ** where gk = (0 gk / OCb2s)T,p • We further define

j = 1 2]~kgk*' + (1-~s)]~qbkgk*" (42) r o ( 1 - ~ s ) k k

Ji - g i + ( 1 - ~ s ) i k i (r i~Pi) -1 (43)

Substitution of eqs.(40-43) into eq.(38) yields

[ O } 1¢-1) V (~:-1) . . . O Oc-1)

/+J+J J + J 2 + J 2 + k 2 . . J + J 2 + J n : . . . . .

[ J + J n + J 1 J + J n + J 2 . . . J + J n + J n + ~ , n

After expansion of the determinant, we obtain the analytical expression.

D 0c) = D ~c-i) k11D (2) _ (l-ik i )kll{(,~,])i(~-l)k/--l)[ s_y_.,JiEk[-1+J l(a+~,ji ki--l)] i i i i

_ ( ~ D i O C - 1 ) j i k i - - 1 ) ( l + ~ j i ~,/-1 j 1Ekt..- i)} (45) i i i

w h e r e D (2) is the second-order stabil ity criterion, i.e., the sp inodal criterion.

G l l

D(2) = G21

On 1

- J + 2 J l + k l

J + J 2 +J 1

J + J n + J 1

G12

0 2 2 " • "

n2 " " "

J + J l + J 2

J + J 2 + J 2 + k 2

J + J n + J 2

O ln

O 2n

G n n

• . . J + J l + J n

• . . J + J 2 + J n

• . . J + J n + J n + k n

The corresponding analytic version of this determinant is

D (2) = i - i ~ i [ (1%~/i k/--l) 2 + ~[~k / - l ( J -~ / / i2~ / -1 ) ] i i i i

(44)

(46)

(47)

C r i t e r i o n f o r S p i n o d a l s

The spinodal is the boundary between the metastable and the unstable regions which satisfies the condition that the second-order stability criterion equals zero, D (2) _- 0. Because I'Iki is always positive, we then define the spinodal criterion:

i

Y. Hu et al. / Fhdd Phase Equilibria 104 (1995) 229-252

Fs p = (14-~_aji ~./-1)2 + ]~ , t . - - l ( j_a~aj i2~ , / -1) _ 0 i i i

239

(48)

Criterion for Critical Points

For a critical point, D (2) - - 0 must be satisfied and, in addition, D (3) must equal zero. To use eq.(45), we first derive D~ 2) from eq.(47).

Ok(2) ffi 20(2) / ~ k ffi 1-I~'i [ 2(14-*~-aJi~'/-1)(a~-r/i'~'/-1) 4- ]P~,zl(J*-2,~.¢liJi')~71) i i i i i

+ 2(1+~.,/i)~71)Jkr~ + (J-~cli2~.:~l)rk + ~.,)~71(J~'-j2r~)] (49) i i i

w h e r e Ji "ffi ~Ji / ~ k , J" ffi oJ / ~ k - J k " T h e f o r m e r depends o n i b e c a u s e

gi and gi in Ji are functions of r i. The latter is independent of k because of cancellation.

Because 2i is always positive and D (2) - 0, we then define the {...} term in eq.(45) as the critical-point criterion. Substitution of eq.(49) yields

l%~,J i X71

Fcr = ~__ji3ri ~,~-1 + j , _ 3 ~ j i J i ,~,?1 _ 3(~j i2ri ~?l_j~_ji ,~/-1) i i i i i

i

l+~.ffi ~/--I 14-~fli ~,/-1

4" 3~_¢1 i r i ~.ZI( i .)2 _ ~.f i ~./-1( i ) 3 ffi 0 ( 5 0 )

i i

Criteria for Multiple Critical Points

For a higher-order multiple critical point, we use eq.(38). For simplicity, we suggest that the Flory parameter gi is independent of chain length. Eq.(45) can be simplified as D ('¢) -- - ( l"I~i )~.11J~.,Di('c-1)ZTk We then define the criterion for the

i i (m-2) th order critical point by the conditions for the m th order stability.

Fcr(m_2) ffi I ' I ~ , / - l ~ p i ( ~ - l ) ~ , i --1 -- 0 , K = m (51) i i

Here we include I'Iiv/-1 b e c a u s e Di(~-I) has a term 1-Iki, as indicated in eq.(49). i i

We then have the following criteria for multiple critical points when the Flory parameter g depends only on the total polymer volume fraction:

1 2g** + 2(1-2@s )g ** (1) Fcr (0) ffi Fsp - ro ( l _ ~ s )

+ ~s(1-- '~s)g **(2) + ~ = 0 (52) (I)sV 1


1 F c r ( l ) -

r o (l_qbs)2 -6g ** (1)+3(1-2~ s )g ** (2)

+ dPs(1._.~s)g**(3 ) 1 v2 = 0 CI)2V ? V 1

2 12g **(2)+4(1-2~ s )g ** (3) Fcr (2) = ro ( l _ ~ s ) 3

**(4). 1 , 3V2 V3

+dPs( l" -dPs)g + ~ 3 v 3 ~ v~ vl ) = 0

20g ** (3)+5 ( 1 - 2 ~ s ) g ** (4) 6 Fcr (3) =

r o ( 1 - ~ s ) 4

(53

(54)

15v2 3 10v3v 2 1 ( _ _ _ _ + v 4 ) _ 0 (55) + d ~ s ( 1 - ~ s ) g * * ( 5 ) - 4 4

.svl where g**(k) is t he kth-order derivative of g with respect to qb s, g**(k)__ Okg** / O ~ s ~, g**(1) and g (2) are the same as g " and g**". The

kth-order moment of chain length v k is defined by v k = ~ ~ i rk [ d~ s . Subscripts i

cr(1), cr(2), cr(3) denote critical point, double critical point and triple critical point, respectively. Relations between moments of chain length and various average chain lengths are : r n = 1 / v_ 1 , r w = v 1 , r z = v 2 / V 1 , r z + m ---- Vm+ 2 [ Vm+l ,

where rn , r w and r z are the number average, weight average and z average chain lengths, respectively. Eqs.(52-55) are consistent with those derived by ~olc, Kleintjens and Koningsveld [6] using a series expansion method.

Multiple critical points exist only when we can obtain meaningful higher-order derivatives for the Flory .p, arameter g as shown in eqs.(52-55). For the simple Flory-Huggins theory, g is a constant independent of polymer concentration. Therefore, only ordinary critical points can be predicted by eqs.(52) and (53). How- ever, experiments and modern theories show that the Flory parameter g depends strongly on polymer concentration. The generalized model presented here gives a reasonable concentration-dependence for g . Because of its theoretical basis, the revised Freed model can be used to calculate meaningful higher-order derivatives and therefore, higher-order multiple critical points.

A D O P T I N G A C O N T I N U O U S D I S T R I B U T I O N F U N C T I O N

Composit ions of polydisperse polymers are usually expressed by a continuous distribution function W (I), where I is a distribution variable such as chain length r or relative molar mass M. Distribution function W ( I ) satisfies the following nor- malization constraint: o o o o

f W ( 1 ) d l = 1 , f ~ s W ( 1 ) d l = • s (56) 0 0

1I. Hu et al. / Fluid Phase Equilibria 104 (1995) 229-252 241

In phase-equilibrium calculations including criteria of various orders, we need various summations and moments, such as M 0 in eq.(37), j~.g[mJi 'n ~.i --1 in eqs.(48)

i and (50), and v k in eqs.(52-55). They are calculated using the continuous distribu-

o o

tion function. For example, M o "~ ~-aKi d~i/di)s z I g ( I )W ( I )d l , and i 0

OO

vk "~ ]~ ~ i rk [ ~s "~ S r k ( 1 ) W ( l ) dl , where K(1) and r(1) are counterparts of K i i 0

and r i , respectively.

In polymer science, it is customary to use the gamma distribution function for W(1):

W(1) -~ (l-7)(a-1) exp(/--~--~) (57) [3ar(c0 P

where or, [3 and ~, are three distribution parameters; F is the gamma function. For polymers, parameter ~/ is usually set equal to zero. Using the g~mma distribution often leads to convenient calculations. However, when a polydisperse polymer has an arbitrary distribution, it is better to use a derivative method based on a spline fit, as discussed previously (Ying, Ye and Hu, 1989).

E L L U S T R A T I V E E X A M P L E S

We use the revised Freed model (Case 4) and the corresponding lattice-fluid model (Case 5) for illustration. We calculate cloud-point curves, shadow curves, spinodals and critical points under various conditions.

Effect o f Chain-Length Dependendence of Energy Parameter

For the revised Freed model for a close-packed lattice, eq.(16) gives ~(I)-~ e e + er I -d , where the independent distribution variable l-~r (or 1=]14).

Parameter e r measures the chain-length dependence of the energy parameter. When e r equals zero, the energy parameter is independent of chain length. Figure 1 shows coexistence curves and spinodais for a model system with three different values of er: e r / k - - - 3 0 , 0, 30K (d = 1/3). The average value of the energy parameter calculated by eav--Se(1)W(1) dl is kept constant in all three cases, 13av[k z 82.0K. A gamma distribution is used to describe the chain-length dependence of compositions of polydisperse polymer solutes; distribution parameters are (x -- 2.5441 , [3 -- 607.19 , ),-- 0, while r o -- 1.0. Figure 1 shows that the larger er, the lower is the critical temperature and the critical volume fraction of the polymer, the flatter is the cloud-point curve and the shadow curve. As shown in Figure 1, the critical point is not located at the maximum of the cloud-point curve and the shadow curve but at the intersection of the two curves. It has slightly shifted to the region of higher polymer concentration. Figure 2 shows typical distributions for a principal phase ((x) and conjugate phases ([3) corresponding to two different concentrations of the principal phase. In these calculations, e r =0.


305 - - ! c -

. . . . 8 <" f , [ ,

295 cloud point curve W / ",, w([ ~) (qb~ a~ = 0.115) - - shadow curve 22 X 1 0 4 [ i ',

3057 spinodal ! 6 ] I' ',, - / _ I'/I

3005_2_ 4 i I ' / ~ w(a) 29 I," ,...

i ~'""'w([3)(O(~)=OO1) ....... 305 r~ . \ ', o 2 , ' ; . ., / ,//

300t " : i I' /

0 0.04 0.08 0.12 0.16 0 2 4 6 {I)S r x 10 -3

Figure 1. Effect o f chain-length dependence o f energy parameter on liquid-liquid-

equilibrium coexistence curves using the revised Freed model, e r / k = (1) -30K,

(2) 0, (3) 3OK.

Figure 2. Typical distribution curves for compositions o f polymer solutes in principle

phase and conjugate phase for two values o f ~ ( a ) . S

7

450

T / K

40O

350

300

250 L 0

I

\

(

/

0

, / / 4 /

3

2

0.1 {IOs

5

• J

0.2

Figure 3. Effect o f interaction energy

between solvent and polymer on liquid-

liquid-equilibrium spinodals using the

lattice-fluid model. ~ = ( I ) 0.0300,

(2) 0.0325, (3) 0.0350, (4) 0.0375,

(5) 0.0400.

E Hu et al./Fluid Phase Equilibria 104 (1995) 229-252 243

Effect o f the Interaction Energy Between Solvent and Polymer

In contrast to the close-packed lattice model, interaction energies %0, ~ss and

~os all appear in the relevant equations of the lattice-fluid model, while in the

former case, only the difference, i.e., Eoo +ess-2~os is needed. Interaction energies %0 and ess are pure-component parameters, while eos measures the interaction between the solvent and the polymer segment.

Eo s ---- ( l _ ~ ) ( E o ° Es s)1/2 (58 )

where ~ is a binary parameter. Figure 3 shows the effect of eos on spinodals for a

model system at 10 MPa. Pure-component parameters used are r o -- 4.734, ot = 6,13 ~- 208.3 ; eoo/k -- (167.63+15620.0K/T)K, ess/k - -263.28K. Five different %s are used: ~=0.0300, 0.0325, 0.0350, 0.0375,

0.0400. Figure 3 shows that the larger ~ (i.e., the lower ¢os), the higher is the UCST, the lower is the LCST, the narrower is the gap between the UCST and the LCST. In the extreme case, when ~--0.0400, hour-glass-shaped phase behavior appears. Figure 4 shows some of the corresponding cloud-point curves and shadow c u r v e s .

Effect o f the Binary Size Parameter

Typically, the energy parameter has significant influence on the critical temperature but not on the critical composition. A binary size parameter c r defined by eq.(17), is usually needed to fit the critical composition. Figures 5 and 6 show the effect of c r on coexistence curves for the revised Freed model of a close-packed lattice and on spinodals for the corresponding lattice-fluid model, respectively, Cr=0.1, 0, -0.1 for Figure 5; Cr=0.2, 0.1, 0, -0.1 for Figure 6. For Figure 5, parameters used are the same as those in section 6.2 with ~=0.0375. For Figure 6, distribution parameters are changed, (x--51,[3--24.51. Figures 5 and 6 show that the larger Cr, the higher is the critical volume fraction of the polymer solute. However, unlike the effect of e, the critical temperature shifts at the same time. As shown in Figures 5 and 6, the larger Cr, the lower is the UCST, the higher is the LCST (in the lattice-fluid model), the wider is the gap between the UCST and the LCST. In the extreme case of a smaller Cr, hour-glass-shaped phase behavior appears, as shown in Figure 6, when c r =-0.1.

Effect o f the Polymer's Molar Mass

Figure 7 shows spinodals for model polydisperse polymer solutions with different weight-average chain lengths (molar masses) for the polymer solute, r w = 625, 1250, 2000, 2500. Other parameters used are the same as those in Figure 6 of section 6.3 for a lattice-fluid model. Figure 7 shows that the larger the average chain length (molar mass), the higher is the UCST, the lower is the LCST, the narrower is the gap between the UCST and the LCST. Also, the critical volume fraction of the polymer solute becomes smaller. In the extreme case, when rw=2500 , hour-glass-shaped phase behavior appears.

244 Y. Hu et al. /Fluid Phase Equilibria 104 (1995) 229-252

450

T / K

400

350,

- - c l o u d p o i n t c u r v e

shadow c u r v e

300 3

2 /

~ - ~ ~ - - - - ~ ~ 1 250 /

0 0.1 0.2 ~s

380

T / K

360

340

320

300

0

/ \~ 3 \ ,

- - c l o u d p o i n t c u r v e

shadow c u r v e

2

I

0.1 ( I ) s 0.2

Figure 4. Effect of interaction energy between solvent and polymer on liquid-liquid- equilibrium coexistence curves using the lattice-fluid model. ~ = (1) 0.0300, (2) 0.0325, (3) 0.0350, (4) 0.0375. Figure 5. Effect of binary size parameter on liquid-liquid-equilibrium coexistence curves using the revised Freed model. Cr = (1) 0.1, (2) 0, (3) -0.1

500

T / K

450

400

350

300

250

~ 2

f ~ 2

Figure 6. Effect of binary size parameter on liquid-liquid-equilibrium spinodals using the lattice-fluid model, c r = (1) 0.2,

(2) 0.1, (3) 0, (4) -0.1.

I I

0 0.1 0.2 0.3 ~ s

E Hu et al./Fluid Phase Equilibria 104 (1995) 229-252 245

E f f e c t o f P o l y d i s p e r s i t y

Polydispersity can be expressed by r w [r n or M w [ M n . Distribution parameters for the gamma distribution function are determined by those ratios: ~5 = r w - r n , ot -~ r w / ~5. Figure 8 shows the effect of polydispersity on coexistence curves with r w / r n -- 1.00,1.10,1.50; rw--1250. Other parameters are the same as those in Figure 6 of section 6.3 for a lattice-fluid model. Figure 8 shows that the larger the polydispersity, the more separate is the shadow curve from the cloud-point curve.

E f f e c t o f P r e s s u r e

Figure 9 shows the effect of pressure on spinodals for a model system whose parameters are the same as those in Figure 6 of section 6.3 for a lattice-fluid model. As shown in Figure 9, the lower the pressure, the higher is the UCST, the lower is the LCST, the narrower is the gap between the UCST and the LCST. In the extreme case, when the pressure is near zero, hour-glass-shaped phase behavior appears.

DISCUSSION AND CONCLUSION

We have presented a generalized molecular-thermodynamic model for mixtures with one solvent and one polydisperse polymer using a generalized formal Flory- Huggins theory where the "Flory parameter" depends on polymer concentration and chain length as dictated by any one of several theories. In some cases, volume change upon mixing is taken into account. Almost all existing models, including close-packed lattice models, and lattice-fluid models, can be cast into this generalized Flory-Huggins model with different expressions for the polymer-concentration dependence and chain-length dependence of the Flory-Huggins parameter. Other models (not mentioned here), such as free-volume theories, can alsobe treated in a similar way. Two approximations have been used. In one of them we neglect the inverse of the polymer chain length in some expressions because the chain length of the polymer is always very large compared to that of the solvent. In the other we use the total polymer volume fraction in the energy term instead of individual volume fractions for different polymer components. This approximation is reasonable because the segments in different polymer components are essentially the same; the environment is primarily determined by the total concentration of segments rather than concentrations of individual components. The chain-length dependence of the Flory parameter has been taken into account. The approximation is limited to the concentration dependence. It is only in this concentration dependence, where we use the total volume fraction to replace the individual volume fractions of polymer components.

Based on this generalized model, a generalized continuous-thermodynamic framework has been established which contains generalized expressions for chemical potentials, spinodal criterion, critical-point criterion and criteria for multiple critical points. Different models used are distinguished by different functional forms of the Flory parameter and its derivatives, g , g " and g** '" and g**(k) as


[ q

i

440] \ ~ J

390i

340 !

290/ . . . . . . . . . 0 0.1 0.2

~ s

460[

T / K 2 : 1 3

410 - - c l o u d p o i n t c u r v e

shadow c u r v e

360

310 0

( m o n o d i s p e r s e ) 1

2 ~ 2 3 .... ~: : : i 3

0.1 q~s

1

0.2

Figure 7. Effect of weight-average chain length of the polymer solute on liquid-liquid-

equilibrium spinodals using the lattice-fluid model, r w = (1) 625, (2) 1250, (3) 2000,

(4) 2500. Figure 8. Effect of polydispersity on liquid-liquid-equilibrium coexistence curves using

the lattice-fluid model, r , , / r , = ( 1 ) 1.0, (2) 1.1, (3) 1.5.

F 440 ~',

T /K

390

340

1

2

3

4

3 2 1

Figure 9. Effect of pressure on liquid-liquid- equilibrium spinodals using the lattice-fluid model, p = (1) 10MPa, (2) 5MPa, (3) 2MPa, (4) 0.

2 9 0 " . 0 0.1 03 s 0.2


used in those equations derived from AmixG. A binary energy parameter and a binary size parameter are used to fit experimental critical coordinates. Cloud-point curves, shadow curves, spinodals and critical points have been calculated under various conditions which cover different types of liquid-liquid-equilibrium phase equilibria including UCST, LCST and hour-glass shaped phase diagrams.

If we have experimental critical points, spinodals, cloud-point curves and shadow curves, we can obtain the binary energy parameter and the binary size parameter as well as their temperature dependence if the experimental data cover a significant temperature range. Considering the difficulty of obtaining monodisperse polymer samples, we can now use a polydisperse polymer sample with a known molecular-weight distribution for experimental work (e.g. cloud points) and then use the parameters obtained experimentally to calculate phase-equilibrium properties for the same system but with a different molecular-weight distribution.

Two different procedures, i.e., the discrete multicomponent method and the continuous functional method, have been used for deriving equations. Consistent results have been obtained by both methods. The discrete method is rigorous because polymer components are discrete. A continuous distribution function is only used to calculate various moments in those expressions. The functional method has the merit of mathematical integrity such that the continuous distribution function is built inside the whole framework. While the composition is, in fact, discrete, the continuous distribution provides a good approximation.

The present work is only for solutions containing one solvent and one polydisperse polymer. Work in progress extends the methods discussed here to systems containing mixed solvents or mixed polymers and polymer blends with polydisperse polymers.

ACKNOWLEDGMENT

This work was supported by the Chinese National Science Foundation and the Marshall Laboratory of the E. I. Du Pont de Nemours & Company. Additional support was provided by the Director, Office of Basic Energy Science, Chemical Science Division of the U. S. Department of Energy under Contract No. DE-AC03-76SF00098.

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A P P E N D I X : F U N C T I O N A L M E T H O D

For the functional method, the generalized model is oo oo

Am~,4 *o ! * s W ( 1 ) = - - I n * o + ln[*~W(1)]dl + * o ~ ¢b~W(1)g*(1)dl (AI)

N r kT r o r (1) o

- ! * s W ( I ) * o f * sW(1)g**( l ) dl (A2) Gv - --AmixG = --*° ln ,o + l n [ . sW(1)]d l + N r kT r o r (I) o

where g* (1) and g** (I) are counterparts of gi and gi in the discrete method, respectively. Am~xA and Am~G or G~ are functionals.

Chemical Potentials

For the discrete solvent component o, the chemical potential can be derived directly from eq.(A2).

(P,o -I~o~)/kT = O(Am~ G IkT) [ ON o oo

--- ln(1--~s) + *s(1-ro l rn ) + ro*Z~[ ~ g** (1)W(l)dl - (1-*s) ~ g * " ( l ) W ( 1 ) d l ] (A3) 0 0

where g ** '(1 ) = (Og ** ( l ) /O*s)r 4~"

Chemical potentials for continuous components are defined by

kt(1) = OG / O[N r dP s W (1)Jr (1)] (m4)

where the denominator is physically comparable with ON i in the discrete method. But now it is a variation of a functional. When applying eq.(A4) to eq.(A1), we meet the problem of taking derivatives for an integral with respect to a functional. In the theory of functionals, for an integral qJ = ~ f (Nt)dl , the derivative of v/with respect to the functional Nt, in eq.(A4), N t .~ Nrd; s W(1)/r(1), is defined by

0W Of (N t) = ( A S )

The expression for the chemical potential of a continuous component I is then derived.

[~(I )-~'~(1 )]lkT = O(Am~ G Ikr ) I ~ [Nr *s W (I)lr (1)]

-- l n [*sW( l ) ] + 1 - r(1)lr o + r(1)*s(1-rolrn) /r o + (1-*s)r(1)g** (l)

- r (1)*s (1-*s)[ ~ g** (I÷)W(I÷) all÷ - (1-~s) ~ g ** '(I÷)W (l÷)dl ÷] (A6) 0 0

Eqs.(A3) and (A6) are equivalent to eqs.(31) and (32) in discrete method.

Spinodal Criterion

We need to derive an expression for G2G~ as the spinodal criterion where G~ is defined by eq.(A2).

For deriving a higher-order variation for functionals, Kehlen, R~itzsch and Bergmann (1987) sug- gested

GkOv = OkGv{*o+t~*o , * s W ( l ) + t ~ [ * s W ( l ) ] } / 0t k It=o (A7)

The relation between the two dependent variations b* o and 5[*s W(1)] in eq.(A7) is found by minimizing 5a~ using the Lagrange method of undetermined multipliers.

We first rewrite eq.(A2) for the reduced Gibbs function of mixing per site as a functional of variations ~*o, 5[*s W(1)] and variable t .

oo

qb s W (l)+t 5[*~ W (I)] ln{~s (~v - * o +t ~I% ln(*o +t 8 ~ o ) + ~ W (1)+t 8 [ ~ s W (1)] }dl ro o r(1)

+ (~o+t~J.o) { . s W ( l ) + t S [ . s W ( l ) ] } [ g * * ( l ) + t d g (1) G~o+_t20~g(1)l o de' o 2 O~ 2 (Sqbo)Z]dl (A8)

Substitution of this equation into eq.(A7) for k--2 gives the second-order variation.


(bOo) 2 ~ {8[do, w(1)]} 2 820~ = + J0 d/ ro doo r if)dos W (1)

oo

+ 2 f ~I~o~[do~W(1)][g**(1)--Oog**'(l)]dl - 2 f (bOo)2dosW(1)[g**'(I)-doog**'(1)12 ]dl (A9) 0 0

where g**'(1) = (3g** (1)13dos)r, p , g** "(I) = (32g** (l)/3do2)r#. Variations ~ o and ~[dos W(I)] are not independent because ~z~ v should be a minimum. On the other hand, 5doo and 8[dosW(1)] are also sub- jected to the foUowing constraint:

f 5[dosW(1)]dl = 5dos = -Sdoo (A10) 0

Using this equation as constraint, the relation between two variations can be obtained by seeking the con- ditional extremum for 5:Gv by applying the Lagrange undetermined-multiplier method.

We first take derivative for 5zt~v in eq.(A9) with respect to the variation 5[dos W(1)] . According to eq.(A5), keeping 5do o unchanged, we obtain

3~20v 25[dos W (1)] = + 28doo [g ** (1)-doog ** '(I)1 (A11)

~8[dos W (I)] r (I)dos W (I)

We then introduce undetermined multiplier -2K to eq.(A10). After taking derivative and adding to eq.(A11), by setting equal to zero, we obtain

K 8[dos W (I)] . . . . , = + ~ o [ g (1)"~og (1)] (A12)

r (1)do s W(1)

Substituting this equation into eq.(A10), we can solve for K.

~, = 8doo[d~s(<g r>--doo<g r>)-l]/dosVl (A13)

where <g**r > and <g** ' r> are calculated by the following general equation.

<g**t g** ,ra g** ,,n rk> = fg**t (1)g** "m (1)g** "n (l)rk ( l )W (1)dl (A14) o

Combining eq.(A12) and eq.(A13), we obtain the relation between two variations.

~)[dosW(1)] = (~doo)dosr(1)W(1)[<g r>-doo<g r> 1 .g** (1)+doog**,(l)] (A15) V l d o s V l

Substituting this equation into eq.(A9), we have ** * * , 2 ** ** _ _ (<g r>-doo<g r>) r>-doo<g 'r>

52~v = [ 1 + 1 +do s _2.<g r o d o o V l d o s V 1 V 1

* * 2 ** * * t 2 ** - dos(<g r>-2doo<g g r>+doo<g '2r>) - 2dos(<g**'>-doo<g**">/2)] (Sdoo) 2 (A16)

The spinodal criterion is then established: * * , 2 ** * * ,

Fs p = 62~v 1 1 (<g**r>--doo<g r>) r>-doo<g r> (Sdo o)2 - r o ~ + ~ + dos - 2 <g

V 1 V 1

* * 2 - - - ** * * t - - 2 ** - ws(<g r>--Zq)o<g g r>+q~o<g '2r>) - 2dos(<g**'>-doo<g**">12) = 0 (AI7)

To compare with the corresponding equation in the discrete method, i.e., eq.(48), we set relations between some terms and their corresponding terms in the discrete method as follows:

** • ** $1 roldoo I -- 2dos<g > + doo~s<g > ---> J

-g** ( l) + doog**'(l) --> Ji

I / [ r ( 1 ) ~ W(1)] --~ X~

We then have

~ s v l -- f Osr (1 )W( l )d l = ]~ ~.[q 0 i

(A18)

(A19)

(A20)

(A21)

K Hu et al. / Fluid Phase Equilibria 104 (1995) 229-252 251

* * * * t -~s(<g r>-~o<g r > ) = ~ , J i ~ 7 L (A22)

i

*'2 **g** = ~ s (<g r >-2qb o <g ' r >+qboZ<g** '2r >) ~ji2~./-i (A23) i

Subst i tut ion into eq.(A17) and defining a new Fsp,

FspOpslvl l ---> Fs p (A24)

Eq.(48) in the discrete method is recovered.

Critical-Point Criterion

For a critical point, it should satisfy the condit ion that the third-order variation of the Gibbs function of mixing equals zero. By a procedure similar to that in the derivation for the spinodal criterion, we have

all) s <r2> ** ** 1 <--~>3 ( <g r>--cb°<g ' r>)~ 53~v = [- ro~2

< r 2 ~ ** ** ** ** ** 9 * * 1 3 2 2 t 2 t . 2 3 3 . + - - + (<g r >-3dPo<g g r2>+3dPo<g g r >--gPo<g r >)(I), qb2<r >3

+ - < r 2 > , ** _ ~ < ' r > ) 2 + 3qb s <g**r2>-dPo<g **'r2> * * * * * * j 9 (<g r>-dPo<g r > ) - 3 < r > 3 [<g r > q'o g <r>2 , ,

<g r >-2dPo <g g r >+~o <g '2r2> . . . . , < r 2 > * * ~ * * , * * 2 2 * * * * , 2 2 * *

- 3 _-Z77T~-(<g r>--(po<g r > ) - 3 ~ (<g r>-~o<g r > ) < r > qb s < r >

+ 3 <g r ->-~°<g ' r - > + 3 <g 2r2>-2~°<g g 'r'>+~2<g ' 2 r ->

< r >2dP s < r > * * * * * * * * t , * *

- 6 <g r>-dP°<g "r>(<g**r2>-dPog**'r2>) - 6ePs<g**'r><g r>-dP°<g r > + 6 < g ' r > < r >2 <r > <r >

**,, ** **,, <g r >-CI)o <g r> + 3dPs<g > + 6 ~ s ( < g * * g 'r>-~o<g**'Zr>) + 3~o~s<g r>

< r >

_ 3qbo < g * * " r > 3CboCbs(<g**g**"r>-~o<g**'g**"r>)] ( ~ o ) 3 (A25) < r >

where <g**tg ** ,rag ** ,,n rk> is defined by eq. (Al4) . We then have the critical-point criterion,

5 3 G v

F~r - - 0 (A26) (500)3

To compare with that obtained in discrete approach, we set two further relations, * * t l rolOo 2 - 3qb~<g > ---) J ' (A27)

- 2 g * * ' ( l ) + dPog**"(l) ~ j r (A28)

Subst i tut ing eqs . (A21), (A22), (A23), (A27) and (A28) into eq.(A26), eq.(50) in the discrete approach is recovered.

Multiple Critical Point

For mult iple critical points, we follow the procedure l~roposed by Browarzik, Kehlen, Riitzsch and Bergmann (1990). For the case when the Flory parameter g is independent of chain length, eq.(A15) for the relat ion between two variations becomes

6[qb s W (1)] = - 5cI) o r ( I ) W (1) / v I (A29)

The spinodal cri terion eq.(A17) becomes

1 + 1 2g ** = FsP - r o - ( 1 - ~ b s) - - q b s Vl - + 2(1-2qbs )g ** (l) + ~s(l '~s)g**(2) 0 (A30)

It is the same as eq.(52) in the discrete method.


For the ordinary critical point, we differentiate Fsp.

1 - 6g **O) + 3(1-2dPs)g**(2) + cbs(1-d~s)g**(3)] ~ o DFsp = -[ r o ( l-dP s )2

oo

1_ .. f r ( l )B[¢I,, W (1) ]d/

Substitution of eq.(A29) yields

1 _ 6g**(1) 8F~p = -[ ro ( l_~s)2 + 3(1-2qv~ )g * * (2) + ~ ( 1 _ ~ ) g * * (3)] ~i~o

We then define the critical-point criterion:

8Fsp Fcr(1 ) = ~3dp °

We then recover eq.(53) by the discrete method.

For the double critical point, we differentiate Fcr(1 ~.

12g ** (2) + 4( 1-2qb~ )g ** (3) -I- ( I ) s ( 1-qb )g ** (4)] 8qbo 2 8Fcr ( 1 ) = - - [( 1 --qV s )~

fr2(l)D[~s W(/)]d/ 0

(A31)

1 v 2

~sZv~ v~ ~I~ o (A32)

(A33)

3v2fr (l)8[qb s W(l)]dl 0

+ (A34)

Substitution of eq.(A29) yields

6Fcro) = - [ 2 12g**(2) + 4(l_2qbs )g ** (3 ) + qbs (1..@s )g ** (4)] 6q~o ro (1-~s )3

1 3vg + _--~-~(--~- v~.) ~ o (A35) V 1

We then define the criterion for the double critical point.

8Fcr(1) (A36) Fcr(2) = ~ o

Eq.(54) in the discrete method is then recovered.

Following the same procedure, eq.(55) for the triple-critical-point criterion can also be recovered.

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POLYMER SOLUTIONS - WordPress.com · 2013-12-10 · NllU ELSEVIER Fluid Phase Equilibria 104 (1995) 229-252 CONTINUOUS THERMODYNAMICS FOR POLYDISPERSE POLYMER SOLUTIONS Ying Hu a,